Physics final exam question: standing wave frequency of thermally expanded string

Physics 205A Final Exam, fall semester 2015
Cuesta College, San Luis Obispo, CA

A Physics 205A student builds a standing wave experiment with a string and a mass hanging over a pulley to create tension, and at a certain temperature it has a fundamental frequency of 20.0 Hz. As the temperature increases, discuss why the fundamental frequency will increase. Only consider the thermal expansion of the string. Explain your reasoning using the properties of wave speeds, periodic waves, standing waves, and thermal expansion.

Solution and grading rubric:

• p:
  Correct. Understands that:
  1. the fundamental standing wave frequency depends on the wave speed and the physical length L between nodes (which does not change);
  2. the wave speed depends on tension (which is set by the hanging mass, and does not change) and linear mass density (which does change);
  3. the linear mass density which depends on the total mass of the string (which does not change) and the overall length of the string (which expands due to the increase in temperature); such that the increase in temperature will decrease the linear mass density, which will increase the wave speed, resulting in a higher fundamental standing wave frequency. (For (1), may instead discuss how the wavelength λ remains constant, being twice the distance from the anchor to the pulley.)
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. Typically does not explicitly discuss how L or λ remains constant in (1).
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete. Typically does not explicitly discuss how standing wave frequency depends on wave speed in (1), or conflates node-node distance L with overall (expanding) string length L.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some garbled attempt at applying thermal expansion to the dependent wave speed and standing wave frequency parameters.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. Approach other than applying thermal expansion to the dependent wave speed and standing wave frequency parameters.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.
  

Grading distribution:
Sections 70854, 70855, 73320
Exam code: final7rUk
p: 8 students
r: 13 students
t: 10 students
v: 28 students
x: 10 students
y: 0 students
z: 1 student

A sample "p" response (from student 0048):